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4

Part of 2009 Federal Competition For Advanced Students, P1

Problems(1)

midpoints, feet of altitudes and other midpoints, lead to concurrency

Source: Austrian Federal Competition For Advanced Students 2009, Part 1, p4

8/30/2019
Let D,ED, E, and FF be respectively the midpoints of the sides BC,CABC, CA, and ABAB of △ABC\vartriangle ABC. Let Ha,Hb,HcH_a, H_b, H_c be the feet of perpendiculars from A,B,CA, B, C to the opposite sides, respectively. Let P,Q,RP, Q, R be the midpoints of the HbHc,HcHaH_bH_c, H_cH_a, and HaHbH_aH_b respectively. Prove that PD,QEPD, QE, and RFRF are concurrent.
geometryaltitudesmidpointsconcurrencyconcurrent